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Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

Definitions: Numerical methods are techniques to approximate mathematical procedures (example of a mathematical procedure is an integral).

Approximations are needed because we either cannot solve the procedure analytically (example is the standard normal cumulative distribution function) or because the analytical method is intractable (example is solving a set of a thousand simultaneous linear equations for a thousand unknowns for finding forces in a truss).

Applications: With the advent of the modern high speed electronic digital computers, the numerical methods are successfully applied to study problems in mathematics, engineering, computer science and physical sciences such as biophysics, physics, atmospheric sciences and geo-sciences.

Possible topics include but are not limited to:

  1. Approximation theory, interpolations.
  2. Numerical ODE/PDE.
  3. Root finding algorithm.
  4. Numerical linear algebra, matrix computations.
  5. Discrete integral transform, FFT, etc.
  6. Linear/Non-linear programming, integer optimization.

For questions concerning matrices, please consider adding the tag.

For questions concerning optimization, please consider adding the tag.

For questions concerning Numerical ODE/PDE, please consider adding the // tag.

References:

https://en.wikipedia.org/wiki/Numerical_method

"Numerical Methods for Scientific and Engineering Computation" by M. K. Jain, S.R.K. Iyengar, R. K. Jain

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Compute Function to Full Machine Precision

If we want to evaluate $$f(x)=\frac{e^x-1-x}{x^2}$$ then we have to observe its large relative error as $x\to 0$. My question is that how can we find a method so that we can compute $f(x)$ to full machine precision for all $|x|<1$? So far, I have…
user307937
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A question about recurrence in a Newton's method problem?

The equation $x^3-x=0$ has three roots,$ -1, 0, 1.$ We use the Newton's method to find the roots. And there are three cases (i) If $x_0>1/\sqrt{3}$, the Newton's method will converge to $1$. (ii) If $x_0<-1/\sqrt{3}$, the Newton's method will…
Tony
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Inequality involving floating point numbers

Given $x_i^{*}$ ($i\geq 0$) be positive numbers on the computer, and $\delta$ is a unit round-off error, $x_i^{*} = fl(x_i) = x_i(1+\epsilon_{i})$ with $|\epsilon_{i}|\leq \delta$ ($x_i$ are positive exact numbers). (a) If $P_n = \prod_{i=0}^{n}…
ghjk
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Find constant $a$ that minimizes expression $\sum_{k=0}^{r}\frac{(y_k - a\sin(x_k))^2}{\ln(1+x_k^2)}$

$ E(a) = \displaystyle\sum_{k=0}^{r}\frac{(y_k - a\sin(x_k))^2}{\ln(1+x_k^2)}$ I need to find constant $a$ that minimizes this expression $E(a)$. I'ts long time since I've done calculus so I need some guidance. I think I should start from…
Damaon
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Beginning to Work on the Runge-Kutta Method

I'm starting to play around with techniques in the introductory numerical methods literature, and I am starting to use the Runge-Kutta method for approximating solutiosn for systems of first-order diff eq.'s, and I'm getting a little tripped up on…
Laqreisha
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Discretization of forward gradient and (backward) divergence with Dirichlet or Neumann

I want to refer to Equation. (10) in http://www.ipol.im/pub/art/2014/103/ . We consider one dimension and the vector $u\in \mathbb R^n$. For simplicity, we choose $n=3$. Neumann boundary condition If we assume the Neumann boundary condition $\nabla…
jakeoung
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How to solve the transcendental equation $\frac{\zeta'(\alpha)}{\zeta(\alpha)}=-\frac{1}{n}\sum_{i=1}^n\ln x_i$

How can I solve the transcendental equation: $$\frac{\zeta'(\alpha)}{\zeta(\alpha)}=-\frac{1}{n}\sum_{i=1}^n\ln x_i,$$ where $\displaystyle\zeta(\alpha)=\sum_{m=1}^\infty m^{-\alpha}\;\;\;?$ The values $x_i \in \{1, 2, 3, ..., 63, 64\}$ are given…
jjepsuomi
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How to derive $f^{'''}(x)$ using Taylor expansion?

I tried doing the difference between the taylor series expansions of $f(x+h)$ and $f(x-h)$, but it didn't reflect the answer that I should get using the third order central divided difference. The approximation for the third derivative using a…
John
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Determine the number of correct digits in the number $x$ given its relative error $E_r$

Determine the number of correct digits in the number $x$ given its relative error $E_r$ (a): $x=0.4785, E_r=0.2\times 10^{-2}$ (b) : $x=386.4, E_r=0.3$ (c): $x=86.34, E_r=0.3$ For the problem (b), we have $E_r=0.3<0.5=\frac{1}{2}<\frac{1}{2}…
Warrior
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Numerical approximation WITH step functions

I'm interested in approximating a continuous curve (actually a data trace) with step functions. I know this is very similar to approximating the area under a curve with rectangles, but there is a slight twist. My signal data is a function y(t),…
user959555
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Number of significant digits in approximate value

Actual value of a quantity is $1.354675$ and the numerical approximate is $1.354595$ then what is the number of significant digits? By using absolute error i.e. $\vert\text{true value} - \text{approx.} \vert < 0.5 \cdot 10 ^{-t}$, where $t =…
Ali Jan
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Projects to understand Numerical Methods

I am a physicist by education and I am doing a PhD right now. Although, knowing numerical methods is not essential in my field, it would still help me understand it better. Just reading introductory books on numerical methods is to dry and boring…
feynmanfan
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Family of linear one-step method, $A$-stable

Consider the family of linear one-step methods defined by $$y_n = y_{n-1} + h(\theta f_n + (1 - \theta)f_{n-1})$$ where $0\leq \theta \leq 1$. Relevant definition- A difference method is $A$-stable if its region of absolute stability contains the…
Wolfy
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family of linear one-step methods, $\delta$-damping

This problem follows from here Consider the family of linear one-step methods defined by $$y_n = y_{n-1} + h(\theta f_n + (1 - \theta)f_{n-1})$$ where $0\leq \theta \leq 1$. 4) Consider the test problem $y' = \lambda y$, $y(0) = C$. A method is…
Wolfy
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Rolle's Theorem to Prove $f^{(n)}$ has a zero in interval

(Use Rolle's Theorem) Let $f\in C^n[\alpha, \beta ]$. Suppose that $f$ has a zero of multiplicity $m$ at $\alpha$ and a root of multiplicity $k$ at $\beta$, where $m \geq 1, k \geq 1$ and $m+k-1=n$. Prove that $f^{(n)}$ has at least one zero in…
hungryformath
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